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Astronomy&Astrophysicsmanuscriptno.gparisi c ESO2008 (cid:13) February2,2008 Constraints to Uranus’ Great Collision IV The Origin of Prospero M.GabrielaParisi1,2,3,⋆,GiovanniCarraro4,5,MicheleMaris6,andAdrianBrunini3 8 1 InstitutoArgentinodeRadioastronom´ıa(IAR),C.C.No5,1894VillaElisa,Argentina 0 e-mail:[email protected] 0 2 DepartamentodeAstronom´ıa,UniversidaddeChile,Casilla36-D,Santiago,Chile 2 e-mail:[email protected] n 3 FacultaddeCienciasAstrono´micasyGeof´ısicas,UniversidadNacionaldeLaPlata,Argentina a e-mail:gparisi,[email protected] J 4 EuropeanSouthernObservatory(ESO),AlonsodeCordova3107,Vitacura,Santiago,Chile 0 e-mail:[email protected] 1 5 DipartamentodiAtronomia,Universita´diPadova,VicoloOsservatorio2,I-35122Padova,Italy e-mail:[email protected] ] 6 INAF,OsservatorioAstronomicodiTrieste,ViaG.B.Tiepolo11,I-34131Trieste,Italy h e-mail:[email protected] p - ReceivedJune2007;acceptedJanuary2008 o r ABSTRACT t s a Context. ItiswidelyacceptedthatthelargeobliquityofUranusistheresultofagreattangentialcollision(GC)withanEarthsize [ proto-planet at the end of the accretion process. The impulse imparted by the GC had affected the Uranian satellitesystem. Very recently,nineirregularsatellites(irregulars)havebeendiscoveredaroundUranus.Theirorbitalandphysicalproperties,inparticular 2 thoseoftheirregularProspero,setconstraintsontheGCscenario. v Aims. WeattempttosetconstraintsontheGCscenarioasthecauseofUranus’obliquityaswellasonthemechanismsabletogive 8 origintotheUranianirregulars. 5 Methods. Differentcapturemechanismsforirregularsoperateatdifferentstagesonthegiantplanetsformationprocess.Themech- 2 anismsabletocapturetheuranianirregularsbeforeandaftertheGCareanalysed.AssumingthattheywerecapturedbeforetheGC, 1 wecalculatetheorbitaltransferofthenineirregularsbytheimpulseimpartedbytheGC.Iftheirorbitaltransferresultsdynamically . 1 implausible,theyshouldhaveoriginatedaftertheGC.Wetheninvestigateanddiscussthedissipativemechanisms abletooperate 0 later. 8 Results. Veryfewtransfersexistforfiveoftheirregulars,whichmakestheirexistencebeforetheGChardlyexpected.Inparticular 0 ProsperocouldnotexistatthetimeoftheGC.DifferentcapturemechanismsforProsperoaftertheGCareinvestigated.Gasdrag : by Uranus’envelope and pull-down capture are not plausible mechanisms. Capture of Prospero through a collisionless interaction v seemstobedifficult.TheGCitselfprovidesamechanismofpermanentcapture.However,thecaptureofProsperobytheGCisalow i X probableevent.CatastrophiccollisionscouldbeapossiblemechanismforthebirthofProsperoandtheotherirregularsaftertheGC. Orbitalandphysicalclusteringsshouldthenbeexpected. r a Conclusions. Either Prospero had to originate after the GC or the GC did not occur. In the former case, the mechanism for the originofProsperoaftertheGCremainsanopenquestion.Anobservingprogramabletolookfordynamicalandphysicalfamiliesis mandatory.Inthelattercase,anothertheorytoaccountforUranus’obliquityandtheformationoftheUranianregularsatelliteson theequatorialplaneoftheplanetwouldbeneeded. Keywords.Planetsandsatellites:general–Planetsandsatellites:formation–SolarSystem:general–SolarSystem:formation 1. Introduction Very recently, rich systems of irregularsatellites (hereafter irregulars)of the giant planets have been discovered.Enabled by the use of large-format digital images on ground-based telescopes, new observational data have increased the known population of Jovianirregularsto55(Sheppardetal.2003),theSaturnianpopulationto38(Gladmanetal.2001,Sheppardetal.2005a,2006a) andtheNeptunianpopulationto7(Holmanetal.2004,Sheppardetal.2006b).TheUraniansystemisofparticularinterestsincea populationof9irregulars(namedCaliban,Sycorax,Prospero,Setebos,Stephano,Trinculo,S/2001U2:XXIVFerdinand,S/2001U3: XXIIFranciscoandS/2003U3:XXIIIMargaret)hasbeendiscoveredaroundUranus(Gladmanetal.1998,2000,Kavelaarsetal. 2004,Sheppardetal.2005b).ThediscoveryoftheseobjectsprovidesauniquewindowonprocessesoperatingintheyoungSolar System.IntheparticularcaseofUranus,theirexistencemaycastlightonthemechanismresponsibleforitspeculiarrotationaxis (Parisi&Brunini1997,Bruninietal.2002(hereafterBP02)). Sendoffprintrequeststo:M.GabrielaParisi ⋆ MemberoftheCarreradelInvestigadorCient´ıfico,ConsejoNacionaldeInvestigacionesCient´ıficasyTe´cnicas(CONICET),Argentina 2 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV Irregularsofgiantplanetsarecharacterizedbyeccentric,highlytiltedwithrespectoftheparentplanetequatorialplane,andin somecaseretrograde,orbits.Theseobjectscannothaveformedbycircumplanetaryaccretionastheregularsatellitesbuttheyare likelyproductsofanearlycaptureofprimordialobjectsfromheliocentricorbits,probablyinassociationwithplanetformationitself (Jewitt&Sheppard2005).Itispossibleforanobjectcirclingaboutthesuntobetemporarilytrappedbyaplanet.Intermsofthe classicalthree-bodyproblemthistypeofcapturecanoccurwhentheobjectpassesthroughtheinteriorLagrangianpoint,L ,witha 2 verylowrelativevelocity.But,withoutanyothermechanism,suchacaptureisnotpermanentandtheobjectswilleventuallyreturn toasolarorbitafterseveralorseveralhundredorbitalperiods.Toturnatemporarycaptureintoapermanentonerequiresasource oforbitalenergydissipationandthatparticlescouldremaininsidetheHillspherelongenoughforthecapturetobeeffective. Althoughcurrentlygiant planetshave no efficient mechanismof energydissipation for permanentcapture,at their formation epochseveralmechanismsmayhaveoperated:1)gasdraginthesolarnebulaorinanextended,primordialplanetaryatmosphereor inacircumplanetarydisk(Pollacketal.1979,Cuk&Burns2003),2)pull-downcapturecausedbythemassgrowthand/ororbital expansionof the planetwhich expandsits Hill sphere(Brunini1995, Heppenheimer& Porco1977), 3) collisionlessinteractions betweenamassiveplanetarysatelliteandguestbodies(Tsui1999)orbetweentheplanetandabinaryobject(Agnor&Hamilton 2006), and 4) collisional interaction between two planetesimals passing near the planet or between a planetesimal and a regular satellite. This last mechanism, the so called break-up process, leads to the formation of dynamical groupings (e.g. Colombo & Franklin 1971, Nesvorny et al. 2004). After a break-up the resulting fragments of each progenitor would form a population of irregularswithsimilarsurfacecomposition,i.e.similarcolors,andirregularshapes,i.e.light-curvesofwideamplitude.Significant fluctuationsinthelight-curvesofCaliban(Marisetal.2001)andProspero(Marisetal.2007a)andthetimedependenceobserved inthespectrumofSycorax(Romonetal.2001)suggesttheideaofabreak-upprocessfortheoriginofthesebodies. SeveraltheoriestoaccountforthelargeobliquityofUranushavebeenproposed.Kubo-Oka&Nakazawa(1995),investigated the tidalevolutionof satellite orbitsandexaminedthe possibility thatthe orbitaldecayof a retrogradesatellite leads to the large obliquityofUranus,butthelargemassrequiredforthehypotheticalsatellitemakesthispossibilityveryimplausible.Anasymmetric infallortorquesfromnearbymassconcentrationsduringthecollapseofthemolecularcloudcoreleadingtotheformationofthe SolarSystem,couldtwistthetotalangularmomentumvectoroftheplanetarysystem.Thistwistcouldgeneratetheobliquitiesof theouterplanets(Tremaine1991).Thismodelhasthedisadvantagesthattheouterplanetsmustformbeforetheinfalliscomplete andthattheconditionsfortheeventthatwouldproducethetwistareratherstrict.Themodelitselfisdifficulttobequantitatively tested.Tsiganisetal.(2005)proposedthatthecurrentorbitalarchitectureoftheouterSolarSystemcouldhavebeenproducedfrom aninitiallycompactconfigurationwithJupiterandSaturncrossingthe2:1orbitalresonancebydivergentmigration.Thecrossing led to close encountersamong the giant planets, producinglarge orbital eccentricities and inclinations which were subsequently dampedtothecurrentvaluebygravitationalinteractionswithplanetesimals.Theobliquitychangesduetothechangeintheorbital inclinations. Since the inclinations are damped by planetesimals interactions on timescales much shorter than the timescales for precessionduetothetorquesfromthe Sun,especiallyforUranusandNeptune,the obliquityreturnsto smallvaluesif itissmall beforetheencounters(Hoietal.2007). Large stochastic impacts at the last stage of the planetaryformationprocesshave been proposedas the possible cause of the planetaryobliquities(e.g.Safronov1969). Thelargeobliquityof Uranus(98 ) isusuallyattributedto a greattangentialcollision ◦ (GC)betweentheplanetandanEarth-sizeplanetesimaloccurredattheendoftheepochofaccretion(e.g.,Parisi&Brunini1997, Korycanskyetal.1990).ThecollisionimpartsanimpulsetoUranusandallowspreexistingsatellitesoftheplanettochangetheir orbits. Irregularson orbits with too large semimajor axis escape from the system (Parisi & Brunini 1997), while irregulars with a smallersemimajoraxismaybepushedto outerorinnerorbitsacquiringgreaterorlowereccentricitiesdependingonthe initial orbitalelements,thegeometryoftheimpactandthesatellitepositionatthemomentofimpact.Theorbitsexcitedbythisperturbation mustbeconsistentwiththepresentorbitalconfigurationoftheUranianirregulars(BP02). InanattempttoclarifytheoriginofUranusobliquityandofitsirregulars,weareusinginthisstudythemostupdatedinformation ontheirorbitalandphysicalproperties. InSection2,weimprovethemodeldevelopedinBP02forthefiveUranianirregularsknownatthatepochandextendourstudy tothenewfourUranianirregularsrecentlydiscoveredbyKavelaarsetal.(2004)andSheppardetal.(2005b).Theoriginofthese objectsaftertheGCisdiscussedinSection3,whereseveralmechanismsfortheoriginofProsperoareinvestigated.Thediscussion oftheresultsandtheconclusionsarepresentedinSection4. 2. Transferoftheirregularstotheircurrentorbits: AssumingtheGCscenario,thetransfersofthenineknownirregularstotheircurrentorbitsarecomputedfollowingtheprocedure developedinBP02forthefiveirregularsknownin2002.Wepresentimprovedcalculationsusingamorerealisticcodetocompute theevolutionoftheirregularscurrentorbitaleccentricities. IfthelargeobliquityofUranushasbeentheresultofagianttangentialimpact,theorbitsofpreexistingsatelliteschangeddue totheimpulseimpartedtotheplanetbythecollision.TheangularmomentumandimpulsetransfertotheUraniansystematimpact weremodeledusingtheUranuspresentdayrotationalandorbitalpropertiesasimputparameters(BP02). JustbeforetheGC,thesquareoftheorbitalvelocityν ofapreexistingsatelliteofnegligiblemassisgivenby: 1 2 1 ν2 =Gm , (1) 1 U r − a ! 1 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 3 rbeingthepositionofthesatelliteonitsorbitatthemomentoftheGC,a itsorbitalsemiaxisandm themassofUranusbefore 1 U the impact.The impactormassis m andG is the gravitationalconstant. Afterthe GC, the satellite is transferredto anotherorbit i withsemiaxisa acquiringthefollowingsquareofthevelocity: 2 2 1 ν2 =G(m +m) . (2) 2 U i r − a ! 2 Wesetν2 = A ν2 andν2 = B (1+m/m ) ν2,whereAandBarearbitrarycoefficients(0 < A 1, B > 0),ν beingtheescape 1 e 2 i U e ≤ e velocityatrbeforetheGC. Thesemiaxisofthesatelliteorbitbefore(a )andafter(a )theGCverifythefollowingsimplerelations: 1 2 r r a = , a = . (3) 1 2 2(1 A) 2(1 B) − − If A < Bthena < a .Inthespecialcaseof B = 1,theorbitsareunboundfromthesystem.If A > Bthena > a ,theinitial 1 2 1 2 orbitistransferredtoaninnerorbit.WhenA= B,theorbitalsemiaxisremainsunchanged(a =a ). 1 2 Thepositionrofthesatelliteonitsorbitattheepochoftheimpactmaybeexpressedinthefollowingform: 2G m B A r= U ′− 2, (4) (∆V)2 "√AcosΨ √(B A)+Acos2Ψ# ± ′− withB = B(1+m/m ).Sincestochasticprocessescanonlytakeplaceatverylatestagesinthehistoryofplanetaryaccretion(e.g. ′ i U Lissauer&Safronov1991),theGCisassumedtooccurattheendofUranusformation(e.g.Korycanskyetal.1990).Themassof UranusaftertheGC,(m +m ),istakenasUranus’presentmass.Ψistheanglebetweenν andtheorbitalvelocitychangeimparted i U 1 toUranus∆V.Ananalyticalexpressionfor∆VisderivedinBP02assumingthattheimpactisinelastic(Korycanskyetal.1990)as afunctionofm,theimpactparameterofthecollisionb,thepresentrotationangularvelocityofUranusΩ,thespinangularvelocity i whichUranuswouldhavetodayifthecollisionhadnotoccurredΩ ,andαwhichistheanglebetweenΩandΩ : 0 0 2R2 Ω2 2ΩΩ cosα 1/2 ∆V = U Ω2+ 0 0 , (5) 5b " (1+ mi)2(1+ mi )4 − (1+ mi)(1+ mi )2# mU 3mU mU 3mU R being the present equatorial radius of Uranus. A collision with the core itself was necessary in order to impart the required U additionalmassandangularmomentum(Korycanskyetal.1990).Sincebisanunknownquantity,wetakeitsmostprobablevalue: b=(2/3)R ,whereR isthecoreradiusofUranusatthemomentofcollisionassumedtobe1.8 104km(Korycanskyetal.1990, C C × Bodenheimer&Pollack1986).Theresultshavea smoothdependencewith theimpactormasswhichallowsustotakem 1m i (Parisi&Brunini1997). ∼ ⊕ Theminimumeccentricityoftheorbitsbeforethecollisionisgivenby: e =2(1 A) 1 if A 0.5 , e =1 2(1 A) if A>0.5, (6) 1min 1min − − ≤ − − whiletheminimumeccentricityoftheorbitsafterthecollisionis: e =2(1 B) 1 if B 0.5 , e =1 2(1 B) if B>0.5. (7) 2min 2min − − ≤ − − Theminimumpossiblevalueof∆V (∆V )isobtainedfromEq.(5)foraninitialperiodT =20hrs(T =2π/Ω )andα=70o min 0 0 0 (BP02).Therefore,althoughsimulationsofsolidaccretionproduceingeneralrandomspinorientations(e.g.Chambers2001),we further assume T = 20 hrs and α=70o in order to set a maximum boundon Eq. (4). Upper boundsin a (a ) and a (a ) are 0 1 1M 2 2M obtainedfromEq.(3)throughEqs.(4)and (5)taking∆V=∆V withΨ=180o,i.e.,assumingtheimpactinthedirectionopposite min totheorbitalmotionofthesatellitesandtakingthepositivesignofthesquarerootinEq.(4): G m (B A)2 G m (B A)2 a = U ′− , a = U ′− . (8) 1M 2M (∆V )2(1 A)(√B √A)2 (∆V )2(1 B)(√B √A)2 min − ′− min − ′− ForeachA,wecalculatethevalueofB(B= B/(1+m/m ))correspondingto thetransfertoa = a,whereaisthepresent ′ i U 2M orbitalsemiaxis of each one of the Uranian irregularsshown in the second columnof Table 1 in units of R . From Eq. (7), this U valueofBprovidestheminimumpossiblevalueofe ,e ,thattheorbitofeachirregularmayacquireatimpactforeachinitial 2min 2m conditionAandeveryinitialconditionforT ,α,Ψandm,i.e.,ifatransferofagivenorbit(A,B)isnotpossibleforΨ=180o,T = 0 i 0 20hrsandα= 70o,thesametransfer(A,B)isnotpossibleforanyotherincidentdirectionoftheimpactorandforanyothervalue ofT andαeither. 0 Since the orbits of the irregulars are time dependent, the orbital evolution of the five Uranian irregulars known in 2002 was computed in PB02 by numerical integration of the equations of the elliptical restricted three body problem formed by the Sun, Uranusandthesatellite.Inthispaper,wepresenttheorbitalevolutionofthenineknownUranianirregularsfor105 yrsusingthe 4 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV Satellite r [km] a[R ] e a[R ] e (∆a)/a) (∆e/e) s U mean i U i i i Caliban 49 283 0.191 287.5 0.1973 1.602x10 2 3.289x10 2 − − Sycorax 95 482 0.541 485 0.5436 6.307x10 3 4.795x10 3 − − Prospero 15 645 0.432 648 0.4342 4.585x10 3 4.997x10 3 − − Setebos 15 694 0.581 701.8 0.5853 1.123x10 2 7.366x10 3 − − Stephano 10 314 0.251 333 0.2781 6.058x10 2 0.1080 − Trinculo 5 336 0.218 361.6 0.2501 7.623x10 2 0.1475 − Ferdinand 6 813 0.660 839.3 0.6701 3.241x10 2 1.533x10 2 − − Francisco 6 169 0.142 280.6 0.3593 0.6604 1.530 Margaret 5.5 579 0.633 649 0.6705 0.1209 5.917x10 2 − Table 1. PresentparametersoftheUranianirregularsandorbitaldampingdueto gasdragexertedbyUranusextendedenvelope. r andaarethepresentphysicalradiusandthepresentorbitalsemiaxisoftheirregulars.e istheircalculatedmeaneccentricity s mean tabulatedinTable2.a ande aretheorbitalsemiaxisandeccentricityjustaftertheGC,while(∆a)/a)and(∆e/e)arethedamping i i i i oftheseorbitalelementssincetheepochoftheGCuntilthecontractionofUranusenvelope. Satellite e e e mean max min Caliban 0.191 0.315 0.072 Sycorax 0.514 0.594 0.438 Prospero 0.432 0.571 0.305 Setebos 0.581 0.704 0.463 Stephano 0.251 0.381 0.121 Trinculo 0.218 0.237 0.200 Ferdinand 0.660 0.970 0.393 Francisco 0.142 0.187 0.093 Margaret 0.633 0.854 0.430 Table2.VariationoftheeccentricityoftheUranianirregularsduetoSolarandgiantplanetperturbationsoveraperiodof105yrs. symplecticintegratorofWisdom&Holman(1991),wheretheperturbationsoftheSun,Jupiter,SaturnandNeptuneareincluded. Themean(e ),maximum(e )andminimum(e )eccentricitiesareshowninTable2foralltheknownUranianirregulars. mean max min Thetransferofeachsatellite fromitsoriginalorbittothepresentoneispossibleonlyforthosevaluesof(A,B)whichsatisfy theconditione <e .Asatellite didnotexistbeforetheimpactifithasnotransferandthesatelliteswiththewidestrangeof 2m max transfersarethosewiththehighestprobabilityofexistingbeforetheimpact. Thetransferswithinarangeof20R aroundeachpresentsatellitesemiaxis(a =a 20R ;atakenfromTable1)forallthe U 2M U ± Uranianirregularsare shown in Fig.1. Thereare few transfersforSetebos, Ferdinandand Margaret.Thismakesthe existenceof thesesatellitesbeforecollisionlittleprobable.TheonlytransfersforTrinculoandProsperoareclosetothepericenterofaneccentric initialouterorbit(e >0.58forProsperoande >0.62forTrinculo).Theminimumeccentricityaftercollisione forTrinculois 1m 1m 2m intherange[0.16-0.23],veryclosetoe (0.237).ThisresultgivesaverylowprobabilityfortheexistenceofTrinculobeforethe max GC.ForProsperoe isintherange[0.52-0.57],e e (0.571).Thereforethissatellite couldnotexistbeforetheGC.Ifthe 2m 2m max ∼ presentlargeobliquityofUranuswascausedbyalargeimpactattheendofitsformation,Prosperohadtooriginateaftertheevent. RelatingtheoriginoftheouterUraniansystemtoacommonformationprocess,alltheUranianirregularsprobablywereoriginated aftertheGC.Thepossiblepost-GCoriginofProsperoandtheotherUranianirregularsisdiscussedinthefollowingsection. 3. OriginofProsperoaftertheGreatCollision In this section, we analyze the possibility that Prospero have been capturedafter the GC. We investigatethe possible dissipative mechanisms able to produce its permanentcapture taking into account that the giant impact is assumed to have occurred at late stagesintheplanetaryaccretionprocess. 3.1.GasdragbyUranus’envelope Bodenheimer&Pollack(1986)andPollacketal.(1996)studiedtheformationofthegiantplanetsbyaccretionofsolidsandgas. Intheirmodel,thesocalledcoreinstabilityscenario,whenthemassofthecoreoftheplanethasgrownenoughagaseousenvelope beginsto formaroundit. For Uranus, its envelopeextendeduntilits accretionradiuswhich was 500R at the end of Uranus’ U ∼ formation (Bodenheimer& Pollack 1986). The formationof Uranus is completed when there is no more nebular gas to accrete. Otherwise, gas accretion by proto-Uranuswould have continued towards the runaway gas accretion phase and the planet would have now a massive gaseousenvelope.Bodenheimer& Pollack (1986) obtained that after the end of accretion, the radius of the envelopeofproto-Uranusremainedalmostconstant( 500R )overatimescaleof104 yrsandthencontractedrapidlyto 8R U U in105yrs.Thefinalcontractiontothepresent-daypla∼netaryradiusoccurredonaslowertimescaleof108yrs. ∼ Korycansky et al. (1990) carried out hydrodynamical calculations of the GC for a large set of initial conditions at the end of accretion. They found a sharp transition between the cases where almost all the mass of the envelope of Uranus remained after the impact and those where it was almost entirely dispersed by the impact. This implies that the impact should have not M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 5 Fig.1.ThetransferscapableofproducingthepresentorbitsoftheUranianIrregulars.A(B)isthesquareoftheratioofthesatellite’s speed just before(after)the impactto the escape velocityat the satellite’s locationjust before(after)the impact.e (e ) is the 1m 2m minimum eccentricity of the orbits before (after) collision. The full-black line A=B divides the upper region (the current orbits arise frominnerorbits)andthe lowerregion(thecurrentorbitsarise fromouterorbits).Thevalueofe tabulatedin Table2 is max shownonthedashedlineforcomparationwithe .Trinculo:emptydowntriangles,Caliban:fullcircles,Sycorax:emptyrhombus, 2m Ferdinand:emptysquares,Francisco:fulldowntriangles,Margaret:emptyuptriangles,Prospero:fulluptriangles,Setebos:empty hexagons,Stephano:emptycircles. dispersedtheenvelope,astherewouldhavebeennonebulargastore-accreteontheplanet.Theyshowedthattheenvelopereacts hydrodynamicallyatimpactanditexpandsoutward.Aftertheshockthegasfalls backonthe coreovera timescale offewhours beingthefinalresultareadjustmentinsteadofacatastrophictransformation.Thetimescaleforthishydrodynamicalprocessismuch shorterthantheorbitalperiodoftheirregularswhichisoftheorderofyears.WemaythenassumethattheGCdidnotchangethe envelopedensityprofile. 6 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV TheextendedenvelopeofUranuscouldbeinprinciple,asourceofgasallowingthecaptureofProsperoandtheotherirregulars aftertheGC.AssumingthattheGCdidnotchangetheenvelopeprofile,wefitfromFig.1ofKorycanskyetal.(1990)thedensity profile of Uranus’ gaseous envelope before the GC, ρ = c R 4 g cm 3 with c = 1036 and R being measured in cm. It gives a g te − − te nebulardensityof 4 10 13gcm 3attheboundaryof500R inagreementwiththeminimummassnebulamodel. − − U ∼ × As a firstapproximation,we computetheratio ofgasmasstraversedby a bodyofdensityρ andradiusr in a characteristic s s orbitalperiodP,tothemassofthebody.AssumingforthebodyacircularorbitofradiusR,wecalculatetheso-calledβparameter (Pollacketal.1979): P 2πRρ πr2 3πρ R β= = g s = g , (9) τ 43πρsr3s 2ρsrs whereτisthecharacteristictimescaleforchanginganyoftheorbitalparameters.Forthepermanentcapturetooccurβcannotbe verysmall,β 0.04(Pollacketal.1979).UsingEq.(9)andassuminganebuladensity10timesthatoftheminimumnebulamodel (c =1037)for≥anobjectthesizeofProspero(ρ =1.5gcm 3)andatProspero’spericenter(R=278R ),β 9 10 5,whichistoo te s − U − ∼ × smalltoaffecttheorbitofProspero. FollowingBP02,wenowinvestigatewithmoredetailthepossibleeffectofgasdragontheUranianirregularsaftertheGCdue toUranus’extendedenvelopebeforeitscontractiontoitspresentstate.FollowingtheprocedureofAdachietal.(1976),weobtain thetimevariationsoftheeccentricityeandsemiaxisaofeachUranianirregular.Thedragforceperunitmassisexpressedinthe form: C πr2 F = Cρ v2 ,C = D s, (10) − g rel 2m wherev istherelativevelocityofthesatellitewithrespecttothegas.Incomputingthesatellitemassm,asatellitemeandensity rel ρ of1.5gcm 3 istakenforallthesatellites(http://ssd.jpl.nasa.gov/?sat phys par).ThedragcoefficientC is 1andr iseach s − D s ∼ satelliteradiustakenforeachsatellitefromTable1. Assuming that the orbital elements are constant within one Keplerian period (the variations of a and e are very small), we considertheratesofchangeoftheelementsaveragedoveroneperiod,thatis: hddati=−Cπ(G(mi+mU)a)21 Z 2π ρg(e2(1++1e+co2seθc)o2sθ)32dθ 0 de = C(1 e2) G(mi+mU) 21 2π ρg(e+cosθ)(e2+1+2ecosθ)12dθ. (11) hdti −π − a ! Z (1+ecosθ)2 0 Sinceaftertheendofaccretionthegasdensityintheouterregionsoftheenvelopecontractsrapidly,wehaveintegratedEqs.(11) back in time on 104 yrsfor the 9 Uranianirregulars.We have taken v as the satellite orbitalvelocity since we have assumed a rel nullgasvelocity.Thisassumptionmaximizestheorbitaldampingforretrogradesatelliteswhichallowsustosetupperboundsin the damping effect for the orbital eccentricity and semiaxis of all the retrograde irregulars. The mean eccentricity e and the mean actual semiaxis a from Table 1 were taken as the initial conditions for the integrations. We take ρ = c R 4 g cm 3 with c = g te − − te 1036 and R = a(1 e2)/(1+ecosθ), a beingmeasured in cm. The orbitaldampingis shown in Table 1, where a and e are the i i − initialsemiaxisandeccentricityjustaftertheGCattheendofaccretionand∆a=(a a)and∆e=(e e ),arethedampingin i i mean − − theorbitalsemiaxisandeccentricity.Stephano,TrinculoandMargarethadexperiencedlittleorbitalevolutionwhileFranciscohad sufferedalargeorbitaldamping(seeTable1).ThepermittedtransfersofFig.1wouldincreaseforFranciscosincetheconditione 2m <e shouldbethensatisfied.However,theorbitalevolutionofthelargersatellites,inparticularofProspero,resultsnegligible.Even i increasingthenebuladensitybyafactorof10(c =1037)thedampingoftheorbitalelementsofProsperoistoosmallwith∆a/a= te i 0.046and∆e/e=0.048.ThissatellitehadnotexperiencedanyorbitalevolutionduetoUranus’extendedenvelopeandthencould i nothavebeencapturebygasdragaftertheGC. The pressure forces acting on a body traveling through the gas not only decelerates it, but also subjects it to stresses. If the stress isgreaterthan thestrengthof the body,the bodyisfracturedin fragmentsofdifferentsize. Thefragmentsmoveaway one anothersincedragforcesvaryinverselywith sizeandactto separatethem.Theaveragepressureontheforwardhemisphereofa non-rotating,sphericalbodyasitmovesthroughthegaswithrelativevelocityv isapproximatelyequaltothedynamicpressure, rel p = 1ρ v2 (Pollacketal.1979). Thebodywillfragmentinto piecesif p Q, where Q is thecompressivestrength.Values dyn 2 g rel dyn ≥ ofQontheorderof3 106 dynecm 2 areneededtoshatterstrong(e.g.rock/ice)targets(whichis10timeslowerthanthevalue − adoptedforasteroids),×whilecompressivestrengthontheorderof3 104dynecm 2areappropriateforrelativelyweak(snow-like) − × targets(Farinella&Davis1996,Stern1996).ForabodyoncircularorbitatthepresentpericenterofProspero(R=278R ),where U v isthecircularspeedaroundUranusatRandtakingc =1037forρ ,p 0.17dynecm 2,andatSycorax’spericenterp 1 rel te g dyn − dyn ∼ ∼ dynecm 2.Inbothcases p <<QandProsperocouldnothavebeenoriginatedbythedynamicalruptureofaparentobject. − dyn Acollisionmayfracturetheparentbodybutiftheenergyatimpactisnotsufficienttodispersethefragments,dragforcesmay acttoseparatethemagainsttheirmutualattraction.Therelativeimportanceoftheseeffectsismeasuredbytheratioǫ ofthedrag j forceonagivenfragmentjtothegravitationalforceactingonjbytheotherfragmentsi(Pollacketal.1979): F m Dj j ǫ = , (12) j F m Xi,i,j Gij j M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 7 whereF isthegravitationalforcebetweentheparticlesiandj,andF isthedragforceontheparticlejofradiusr : Gij Dj j C ρ πr2v2 F = D g j , (13) Dj 8πρ r3 3 s j Thefragmentjisdispersedbythegasifǫ >1(Pollacketal.1979).Inordertoestimatetheorderofmagnitudeoftheeffect,we j computedEq.(12)forj=ProsperoatProspero’spericentertakingintoaccountthegravitationalattractionduetoanotherfragmentof equalsize,andatSycoraxpericentertakingintoaccountthegravitationalforceofSycoraxonProspero.Inthefirstcase,weobtain ǫ =10 5andinthesecondcaseatSycoraxpericenterǫ =0.04.Wethenconcludethatpressureforcesarenotstrongenoughneither j − j tofragmentapossibleparentobjectfromwhichProsperooriginated,nortodisperseitsfragments.InadditionProsperosufferedno orbitalevolutionduetogasdragandcouldnothavebeencapturebyUranus’envelopeaftertheGC. 3.2.Pull-downcapture Within the GC scenario, runaway of the cores of the planets occurred during the first stages of accretion but stopped for each embryoafteritreachedasizeofabout1000Km.At10-35AUthefinalmassdistributioncontainedseveralhundredsofMars-size (or larger) bodies dominating the mass of the residual disk. Beauge´ et al. (2002), investigated the effects of the post-formation planetarymigrationonsatellitesorbits.Theyobtainedthatifthelarge-bodycomponent(composedofMars-sizebodies)dominated themassoftheresidualdisk,thepresentlyacceptedchangeintheorbitofUranusof 3AU istoolargeanditisnotcompatible ∼ withtheobserveddistributionofitssatellites.Evenanorbitalchangeof 1.5AUalreadycausessufficientinstabilitiestoejectall ∼ theUranianirregulars.Pull-downcapturecausedbytheorbitalexpansionoftheplanetcouldthennotbeaplausiblemechanismfor theoriginofProsperoandtheotherirregulars.Pull-downcapturecausedbythemassgrowthoftheplanetaftertheGCwouldnot bepossiblegiventheimpactisassumedtohaveoccurredattheendoftheaccretionprocesswhentherewas nomoremasstobe accretedbytheplanet. 3.3.Collisionlessinteractions Withintheframeworkoftherestrictedthree-bodyproblem,acaptureisalwaysfollowedbyanescape.Toendupwithalongterm capture,thesatellite hastodissipate energyinashorttime.Theentranceenergy∆E withinthe gravitationalfield oftheplanetis (Tsui1999): GM ∆E = 2.15µ2/3(1 δ) ⊙, (14) − − a p µ= M /M andδ<<1,whereM anda arethemassandorbitalsemiaxisoftheplanet. p p p Tsui (⊙1999), suggested a permanentcapture mechanism where a guest satellite encounterssome existing inner orbit massive planetary satellite causing its velocity vector to be deflected keeping the irregular in orbit around the planet. In this way, the effectivetwo-bodypotentialwouldbe abouttwice the entranceenergy∆E ofthe guestsatellite. TheradiusR ofthe orbitofthe 1 guestsatelliteafterdeflectionisthengivenby: 0.23 R = µ1/3a . (15) 1 1 δ p − In the case of Uranus,for a minimumentranceenergyof δ=0,the minimumpermanentorbitalradiusof the guestsatellite is R =955R .ThisvalueofR ismuchlargerthanthepresentsemiaxisofProspero(seeTable1),makingthecaptureofProsperoby 1 U 1 thismechanismimplausible. Thefactthatbinarieshaverecentlybeendiscoveredinnearlyallthesolarsystem’ssmall-bodyreservoirssuggeststhatbinary- planetgravitationalencounterscouldbring a possible mechanismfor irregularscapture (Agnor& Hamilton 2006). One possible outcomeofgravitationalencountersbetweenabinarysystemandaplanetisanexchangereaction,whereonememberofthebinary isexpelledandtheotherremainsboundtotheplanet.Tsui(1999)extendedthescenariooflargeanglesatellite-satellitescatteringto theformationofthePluto-CharonpairassumingthatPlutowasasatelliteofNeptuneandthatCharonwasaguestsatellite.Through Eqs.(14)and(15),theconditionsfortheescapeofthepairwasfound.Followingtheirscenario,letusconsiderthehypothesisthat ProsperowasamemberofaguestbinaryenteringUranus’field,withenergydensity∆E ,abovetheminimumdensitygivenby bin Eq.(14) and δ=0. A close encounter with Uranus could result in disruption of the binary, leading to the ejection of one member andcaptureoftheother.TheminimumsemiaxisR isgivenbyEq.(15).However,eventhisscenarioseemstobeunlikelysincethe 1 semiaxisofProsperoissmallerthan955R . U 3.4.Collisionalinteractions:Break-upprocesses Collisionalinteractionsbetweentwoplanetesimalspassingneartheplanetorbetweenaplanetesimalandaregularsatellite,theso calledbreak-upprocess,leadstotheformationofdynamicalgroupings(e.g.Colombo&Franklin1971,Nesvornyetal.2004).The resultingfragmentsofeachprogenitorbodyafterabreak-upwillformapopulationofirregularsexpectedtohavesimilarsurface composition,i.e.similarcolors,andirregularshapes,i.e.largetemporalvariationsinthelightcurveastheseirregularbodiesrotate. 8 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV Thecriticalrotationperiod(T )atwhichcentripetalaccelerationequalsgravitationalaccelerationforarotatingsphericalobject c is: 3π 1/2 T = , (16) c Gρ ! ob whereGisthegravitationalconstantandρ isthedensityoftheobject.Withρ =0.5,1,1.5and2gcm 3,T =4.7,3.3,2.7and2.3 ob ob − c hrs,respectively.TherotationperiodofProsperoisabout4hrs(Marisetal.2007a),anditseemsunlikelythatlightanddarksurface markingsonasphericalProsperocouldberesponsibleforitslightcurveamplitudeof0.2mag(Marisetal.2007a).Evenatlonger periods,realbodieswillsuffercentripetaldeformationintoasphericalshapes.Foragivendensityandspecificangularmomentum (H), the nature of the deformation depends on the strength of the object. In the limiting case of a strengthless (fluid) body, the equilibriumshapeshavebeenwellstudied(Chandrasekhar1987).ForH 0.304[inunitsof(GM3R )1/2],whereM(kg)isthe sphe ≤ mass of the objectand R is the radiusof an equal-volumesphere,the equilibriumshapes are the oblate Maclaurinspheroids. sphe For 0.304 H 0.390the equilibriumfiguresare triaxial Jacobiellipsoids. Strengthlessobjects with H > 0.390are rotationally ≤ ≤ unstabletofission.TheKuiperBeltobjects(KBOs),beingcomposedofsolidmatter,clearlycannotbestrengthless.However,itis likelythattheinteriorstructuresofthesebodieshavebeenrepeatedlyfracturedbyimpacts,andthattheirmechanicalresponseto appliedrotationalstressisapproximatelyfluid-like.Such“rubblepile”structurehaslongbeenstudiedintheasteroidbelt(Farinella et al.1981) and in the Kuiper Disk (Sheppard& Jewitt 2002, Jewitt & Sheppard 2002, Romanishin & Tegler 1999). Farinella & Davis(1996)obtainedthatKBOslargerthanabout100kmindiameteraremassiveenoughtosurvivecollisionaldisruptionover theageofthesolarsystem,butmayneverthelesshavebeeninternallyfracturedintorubblepiles. Whetheracollisionbetweenanimpactorandatargetresultsingrowthorerosiondependsprimarilyontheenergyoftheimpact and the mass and strength of the target. If the mass of the impactor is small compared to the mass of the target m , the energy s requiredatimpacttoresultinabreak-upisgivenby: 1 3Gm2 m v2 m S + s, (17) 2 s col ≥ s 5γR ms wherev is the collisionspeed, S is the impactstrength,R is the radiusofthe targetand γ is a parameterwhichspecifies the col ms fractionofcollisionalkineticenergythatgoesintofragmentkineticenergyandisestimatedtobe 0.1(Farinella&Davis1996). ∼ The speed of the fragments is critical when the target has a gravity field. Fragments moving slower than the local escape speed re-accumulatetoformrubberpilestructures. WeattempttoinvestigatewhetherProsperocouldbeacollisionalfragmentorifacollisiononaprimaryProsperowouldresult inarubberpilestructure. IncomputingEq.(17),v2 =v2+v2 ,wherev isthescapespeedatthetargetsurfaceandv isthetypicalapproachvelocity col e inf e inf ofthetwoobjectsatadistancelargecomparedwiththeHillsphereofthetarget.FortwobodiescollidingintheKuiperdisk,v is inf givenby(Lissauer&Stewart1993): 5 v2 =v2 e2+i2 , (18) inf k 4 ! wherev isthekeplerianvelocity,eisthemeanorbitaleccentricityandithemeanorbitalinclinationoftheKBOs.Wetake e =2 i k h i hi (Stern1996).Eq.(17)wascomputedusingEq.(18)forvaluesofeintherange[0.01-0.1]andorbitalsemiaxesoftheKBOsinthe range[30-60]AU.Incomputingthemassofthetargetm ,weconsidertheradiusR oftheKBOsintherange[10-500]kmand s ms densitiesbetween[0.5-2]gcm 3.Impactstrengthsintherange[3 104-3 106]ergcm 3weretaken.Weobtainthattargetswith − − × × R 210kmsufferdisruptionforallthevaluesoftheseparametersintheKuiperDisk.Prosperohasaradiusofjustonly15km.If ms ≤ ProsperooriginatedfromtheKuiperBelt,itwouldhavebeenmorelikelyacollisionalfragmentratherthanprimarybody.Prospero wouldpreservean irregularshape afterdisruptionsince it is suchsmall objectthat is unableto turnsphericalbecause its gravity cannotovercomematerialstrength.Prosperocouldhavebeencapturedduringthebreak-upeventifthetwoKBOscollidedwithin Uranus’Hillsphere,whichcouldbepossibleforaminimumorbitaleccentricityoftheoriginalKBOsof0.37. We alsoconsiderthecaseinwhichthetargetisasatelliteofUranuswhichcollideswithaKBO thatenterstheHillsphereof theplanet.Eq.(17)remainsvalidbutthefollowingexpressionofv isconsidered: inf v =v v , (19) inf ip sp − wherev isthevelocityoftheKB0withrespecttoUranusandv isthesatelliteorbitalvelocity,attheepochofthevent.Weassume ip sp thatthesatelliteorbitalvelocityiscircular.Inordertogetboundsintherelativevelocity,wetaketwovaluesofv ,v =v v . inf inf ip sp ± Forv ,weassumethattheKBOdescribesahyperbolicorbitaroundUranusduringtheapproachgiving: ip 2G(m +m ) GM v2 = i U + ⊙, (20) inf a a s kb whereweassumedGM /a astherelativevelocitybetweentheKBOandUranusfarfromtheencounter,(m +m )isthepresent kb i U massofUranusanda t⊙heorbitalsemiaxisofthesatellite.WecalculateEq.(17)usingEqs.(19)and(20)fora intherange[20,60] s kb AUanda [100-700]R forthesamevaluesofS,R anddensitieswehaveusedforthecollisionsamongKBOs.Weobtainthat s U ms M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 9 forallthepossibleparameters,anyUranus’satellitewithradiusR 1000kmsuffersdisruptionifitcollideswithaKBOlarger ms ≤ than10 km.Thisprocesswouldlead to the formationof two clustersofirregulars,one associatedto the preexistingsatellite and theothertotheprimaryKBO.Thisprocesshasthedisadvantagethatitisunlikelythatthepreexistingsatellitewereformedfroma circumplanetarydiskasregularsatellitesgiventhelargeorbitalsemiaxisrequiredforthisobject. Break-upprocessespredictorbitalclustering.However,noobviousdynamicalgroupingsareobservedattheirregularsofUranus. AfurtherintensivesearchofmorefaintirregularsaroundUranusisneededinordertolookfordynamicalandphysicalfamilies. 3.5.TheCGitselfasapossiblecapturemechanism WenowturntothequestionofwhethertheGCitselfcouldhaveprovidedacapturemechanism(BP02).Sinceallthetransferswith A> B leadtoamoreboundorbit,thisprocessmighttransformatemporarycaptureintoapermanentone(seeSection2andFig.1). ′ Moreover,apermanentcapturecouldevenoccurfromanheliocentricorbit(transferswithA=1). ItisinterestingtoestimatethenumberofobjectsN inheliocentricorbitsatthetimeoftheGC,atdistancesfromUranusless thanorequalto300R .AssumethattheGCoccurredwhenUranuswasalmostfullyformed,meaningthatitsfeedingzonewas U alreadydepleted of primordialplanetesimals. We assume that the objectspassing near Uranusat thattime were mainly escapees fromthe Kuiperbelt. Using the impactrate onto Uranusandthe distributionsofvelocitiesanddiametersgivenbyLevisonet al. (2000),andassumingthatthemassinthetransneptunianregionattheendoftheSolarSystemformationwas10timesitspresent mass,a back-of-the-envelopecalculationgivesoneobjectofdiameterD 20kmpassingatadistanceR 300R fromUranus U ≥ ≤ every6yrsattheendofaccretion(BP02).ThetypicalcrossingtimeT amongprotoplanetsintheouterSolarSystemislargerthan C onemillonyears(Zhouetal.2007).ThenumberofobjectspassingnearUranusduringatimescaleT isthen167000,whichgives C a probabilityof 6 10 6 for the capture of an object at about 300 R by the GC. This low rate of incoming objects, makes the − U × possibilityofthecaptureofalltheirregularsfromheliocentricorbitsdifficult.Eventhecaptureofasingleobject,Prospero(note thatProsperocouldnothaveanorbitboundtotheplanetbeforetheGC),turnsouttobelowprobable.Sincetemporarycapturecan lengthenthetimewhichapassingbodycanspendneartheplanet,amoreplausiblesituationarisesifweassumethattheGCcould produce the permanent capture of one or more parent objects which were orbiting temporarily around Uranus being the present irregularstheresultofacollisionalbreak-upoccurringaftertheGC. 4. DiscussionandConclusions ItisusuallybelievedthatthelargeobliquityofUranusistheresultofagreattangentialcollision(GC)withanEarth-sizedproto- planetattheendofthe accretionprocess.We havecalculatedthetransferofangularmomentumandimpulseatimpactandhave shownthatthe GChadstronglyaffectedtheorbitsofUraniansatellites. We calculatethetransferoftheorbitsofthenineknown UranianirregularsbytheGC.Veryfewtransfersexistforfiveofthenineirregulars,makingtheirexistencebeforetheGChardly expected.Inparticular,Prosperocouldnotexistatthetime oftheGC. Then,eitherProsperohadto originateaftertheGC orthe GCdidnotoccur,inwhichcaseanothertheoryabletoexplainUranus’obliquityandtheformationoftheUranianregularsatellites wouldbeneeded.ItisusuallybelievedthattheregularsatellitesofUranushaveaccretedfrommaterialplacedintoorbitbytheGC (Stevensonetal.1986). WithintheGCscenario,severalpossiblemechanismsforthecaptureofProsperoaftertheGCwereinvestigated.IftheUranian irregulars belong to individual captures and relating the origin of the outer uranian system to a common formation process, gas dragby Uranus’envelopeand pull-downcaptureseem to be implausible.Three-bodygravitationalencountersmightbe a source of permanentcapture. However,we foundthat the minimum permanentorbitalradiusof a guestsatellite of Uranusis 955 R U ∼ whilethecurrentsemiaxisofProsperois645R .TheGCitselfcouldprovideamechanismofpermanentcaptureandthecapture U ofProsperocouldhaveoccurredfromaheliocentricorbitasisrequiredwithintheGCscenario,butduetothelowrateofincoming objectsit turnsoutto bedifficult.Break-upprocessescouldbe themechanismforthe originofProsperoandthe otherirregulars in the frame of differentscenarios. Prospero might be a fragmentof a primary KBO fracturedby a collision with anotherKBO. ThefragmentcouldhavebeencapturedbyUranusifthetwoKBOshadaminimumorbitaleccentricityof0.37.Prosperocouldbe a secondarymemberofa collisionalfamilyoriginatedbythecollisionbetweenanothersatellite ofUranusanda KBO wherethe parentsatelliteofProsperocouldhavebeencapturedbyanymechanismbeforeoraftertheGC.Thisprocesshasthedisadvantage thatitisunlikelythatthepreexistingsatellitewereformedfromacircumplanetarydiskasregularsatellitesgiventhelargeorbital semiaxis required for this object. Since collisional scenarios require in general high collision rates, perhaps the irregulars were originally much more numerousthan now. Then, Prospero and also the other irregulars might be the result of mutual collisions amonghypotheticalpreexistingirregulars(Nesvornyetal. 2003, 2007)whichcouldhavebeencapturedbyanyothermechanism beforetheGC. TheknowledgeofthesizeandshapedistributionofirregularsisimportanttoknowtheirrelationtotheprecursorKuiperBelt population.It could bring valuable clues to investigate if they are collisional fragmentsfrom break-upprocesses occuring at the Kuiper Belt and thus has nothing to do with how they were individually captured later by the planet, or if they are collisional fragmentsproducedduringorafterthecaptureevent(Nesvornyetal.2003,2007).ThedifferentialsizedistributionoftheUranian irregularsapproximatesapowerlawwithanexponentq=1.8(Sheppardetal.2005b).Ifweassumethatthesizedistributionofthe nineirregularswithradiigreaterthan7kmextendsdowntoradiiofabout1km,wewouldexpectabout75irregularsofthissizeor larger(Sheppardetal.2005b). ThenucleiofJupiterfamilycometsarewidelyconsideredtobekilometer-sizedfragmentsproducedcollisionallyintheKuiper Belt(Farinella&Davis1996).Jewittetal.(2003)comparedtheshapedistributionofcometarynucleiintheJupiterfamilywiththe shapedistributionofsmallmain-beltasteroidsofsimilarsize(1km-10km)andwiththeshapedistributionoffragmentsproduced 10 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV in laboratoryimpactexperiments.They foundthat while the asteroidsand laboratoryimpactfragmentsshow similar distribution of axis ratio ( b/a 0.7),cometarynucleiare moreelongated( b/a 0.6).Theypredictthatif cometsreflecttheir collisional h i∼ h i∼ originintheKuiperBeltfollowedbysublimation-drivenmasslossonceinsidetheorbitofJupiter,smallKBOsshouldhaveaverage shapesconsistentwith those ofcollisionally producedfragments(i.e., b/a 0.7).To date, constraintson the shapesof onlythe h i∼ largestKBOsareavailable.Prosperobeingabitlargerthancometarynuclei,displaysavariabilityof0.21magintheRband(Maris etal.2007a).Thiscorrespondstoanaxisratioprojectedintotheplaneofthesky,b/aof0.8.Theknowledgeofthesizeandshape distribution of irregulars would shed light in the size and shape distribution of small KBOs as well as on the irregulars capture mechanism. Colors are an important diagnostic tool in attempting to unveil the physical status and the origin of the Uranian irregulars. In particular it would be interesting to assess whether it is possible to define subclasses of irregulars just looking at colors, and comparingcolorsofthesebodieswithcolorsofminorbodiesintheouterSolarSystem.Avalilableliteraturedatashowadispersion in thepublishedvalueslargerthanquotederrorsforeachUranianirregular(Mariset al.2007a, andreferencestherein).We have concludedinMarisetal.(2007a),thattheUranianirregularsareslightlyredbuttheyarenotasredasthereddestKBOs. An intensivesearchforfainter irregularsanda longterm programof observationsableto recoverin a self consistentmanner light-curves,colorsandphaseeffectsinformationsismandatory. Acknowledgements. MGPresearchwassupportedbyInstitutoArgentinodeRadioastronom´ıa,IAR-CONICET,ArgentinaandbyCentrodeAstrof´ısica,Fondode Investigacio´navanzadoenAreasPrioritarias,FONDAPnumber15010003,Chile.MMacknowledgesFONDAPforfinantialsupportduringavisittoUniversidad deChile.PartoftheworkofMMhasbeensupportedbyINAFFFO-FondoRicercaLibera-2006.ABresearchwassupportedbyIALP-CONICET.Weappreciate theusefulsuggestionsbytheReviewer,whichhavehelpedustogreatlyimprovethispaper. 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Preview Constraints to Uranus' Great Collision. IV. The Origin of Prospero

Astronomy&Astrophysicsmanuscriptno.gparisi c ESO2008 (cid:13) February2,2008 Constraints to Uranus’ Great Collision IV The Origin of Prospero M.GabrielaParisi1,2,3,⋆,GiovanniCarraro4,5,MicheleMaris6,andAdrianBrunini3 8 1 InstitutoArgentinodeRadioastronom´ıa(IAR),C.C.No5,1894VillaElisa,Argentina 0 e-mail:[email protected] 0 2 DepartamentodeAstronom´ıa,UniversidaddeChile,Casilla36-D,Santiago,Chile 2 e-mail:[email protected] n 3 FacultaddeCienciasAstrono´micasyGeof´ısicas,UniversidadNacionaldeLaPlata,Argentina a e-mail:gparisi,[email protected] J 4 EuropeanSouthernObservatory(ESO),AlonsodeCordova3107,Vitacura,Santiago,Chile 0 e-mail:[email protected] 1 5 DipartamentodiAtronomia,Universita´diPadova,VicoloOsservatorio2,I-35122Padova,Italy e-mail:[email protected] ] 6 INAF,OsservatorioAstronomicodiTrieste,ViaG.B.Tiepolo11,I-34131Trieste,Italy h e-mail:[email protected] p - ReceivedJune2007;acceptedJanuary2008 o r ABSTRACT t s a Context. ItiswidelyacceptedthatthelargeobliquityofUranusistheresultofagreattangentialcollision(GC)withanEarthsize [ proto-planet at the end of the accretion process. The impulse imparted by the GC had affected the Uranian satellitesystem. Very recently,nineirregularsatellites(irregulars)havebeendiscoveredaroundUranus.Theirorbitalandphysicalproperties,inparticular 2 thoseoftheirregularProspero,setconstraintsontheGCscenario. v Aims. WeattempttosetconstraintsontheGCscenarioasthecauseofUranus’obliquityaswellasonthemechanismsabletogive 8 origintotheUranianirregulars. 5 Methods. Differentcapturemechanismsforirregularsoperateatdifferentstagesonthegiantplanetsformationprocess.Themech- 2 anismsabletocapturetheuranianirregularsbeforeandaftertheGCareanalysed.AssumingthattheywerecapturedbeforetheGC, 1 wecalculatetheorbitaltransferofthenineirregularsbytheimpulseimpartedbytheGC.Iftheirorbitaltransferresultsdynamically . 1 implausible,theyshouldhaveoriginatedaftertheGC.Wetheninvestigateanddiscussthedissipativemechanisms abletooperate 0 later. 8 Results. Veryfewtransfersexistforfiveoftheirregulars,whichmakestheirexistencebeforetheGChardlyexpected.Inparticular 0 ProsperocouldnotexistatthetimeoftheGC.DifferentcapturemechanismsforProsperoaftertheGCareinvestigated.Gasdrag : by Uranus’envelope and pull-down capture are not plausible mechanisms. Capture of Prospero through a collisionless interaction v seemstobedifficult.TheGCitselfprovidesamechanismofpermanentcapture.However,thecaptureofProsperobytheGCisalow i X probableevent.CatastrophiccollisionscouldbeapossiblemechanismforthebirthofProsperoandtheotherirregularsaftertheGC. Orbitalandphysicalclusteringsshouldthenbeexpected. r a Conclusions. Either Prospero had to originate after the GC or the GC did not occur. In the former case, the mechanism for the originofProsperoaftertheGCremainsanopenquestion.Anobservingprogramabletolookfordynamicalandphysicalfamiliesis mandatory.Inthelattercase,anothertheorytoaccountforUranus’obliquityandtheformationoftheUranianregularsatelliteson theequatorialplaneoftheplanetwouldbeneeded. Keywords.Planetsandsatellites:general–Planetsandsatellites:formation–SolarSystem:general–SolarSystem:formation 1. Introduction Very recently, rich systems of irregularsatellites (hereafter irregulars)of the giant planets have been discovered.Enabled by the use of large-format digital images on ground-based telescopes, new observational data have increased the known population of Jovianirregularsto55(Sheppardetal.2003),theSaturnianpopulationto38(Gladmanetal.2001,Sheppardetal.2005a,2006a) andtheNeptunianpopulationto7(Holmanetal.2004,Sheppardetal.2006b).TheUraniansystemisofparticularinterestsincea populationof9irregulars(namedCaliban,Sycorax,Prospero,Setebos,Stephano,Trinculo,S/2001U2:XXIVFerdinand,S/2001U3: XXIIFranciscoandS/2003U3:XXIIIMargaret)hasbeendiscoveredaroundUranus(Gladmanetal.1998,2000,Kavelaarsetal. 2004,Sheppardetal.2005b).ThediscoveryoftheseobjectsprovidesauniquewindowonprocessesoperatingintheyoungSolar System.IntheparticularcaseofUranus,theirexistencemaycastlightonthemechanismresponsibleforitspeculiarrotationaxis (Parisi&Brunini1997,Bruninietal.2002(hereafterBP02)). Sendoffprintrequeststo:M.GabrielaParisi ⋆ MemberoftheCarreradelInvestigadorCient´ıfico,ConsejoNacionaldeInvestigacionesCient´ıficasyTe´cnicas(CONICET),Argentina 2 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV Irregularsofgiantplanetsarecharacterizedbyeccentric,highlytiltedwithrespectoftheparentplanetequatorialplane,andin somecaseretrograde,orbits.Theseobjectscannothaveformedbycircumplanetaryaccretionastheregularsatellitesbuttheyare likelyproductsofanearlycaptureofprimordialobjectsfromheliocentricorbits,probablyinassociationwithplanetformationitself (Jewitt&Sheppard2005).Itispossibleforanobjectcirclingaboutthesuntobetemporarilytrappedbyaplanet.Intermsofthe classicalthree-bodyproblemthistypeofcapturecanoccurwhentheobjectpassesthroughtheinteriorLagrangianpoint,L ,witha 2 verylowrelativevelocity.But,withoutanyothermechanism,suchacaptureisnotpermanentandtheobjectswilleventuallyreturn toasolarorbitafterseveralorseveralhundredorbitalperiods.Toturnatemporarycaptureintoapermanentonerequiresasource oforbitalenergydissipationandthatparticlescouldremaininsidetheHillspherelongenoughforthecapturetobeeffective. Althoughcurrentlygiant planetshave no efficient mechanismof energydissipation for permanentcapture,at their formation epochseveralmechanismsmayhaveoperated:1)gasdraginthesolarnebulaorinanextended,primordialplanetaryatmosphereor inacircumplanetarydisk(Pollacketal.1979,Cuk&Burns2003),2)pull-downcapturecausedbythemassgrowthand/ororbital expansionof the planetwhich expandsits Hill sphere(Brunini1995, Heppenheimer& Porco1977), 3) collisionlessinteractions betweenamassiveplanetarysatelliteandguestbodies(Tsui1999)orbetweentheplanetandabinaryobject(Agnor&Hamilton 2006), and 4) collisional interaction between two planetesimals passing near the planet or between a planetesimal and a regular satellite. This last mechanism, the so called break-up process, leads to the formation of dynamical groupings (e.g. Colombo & Franklin 1971, Nesvorny et al. 2004). After a break-up the resulting fragments of each progenitor would form a population of irregularswithsimilarsurfacecomposition,i.e.similarcolors,andirregularshapes,i.e.light-curvesofwideamplitude.Significant fluctuationsinthelight-curvesofCaliban(Marisetal.2001)andProspero(Marisetal.2007a)andthetimedependenceobserved inthespectrumofSycorax(Romonetal.2001)suggesttheideaofabreak-upprocessfortheoriginofthesebodies. SeveraltheoriestoaccountforthelargeobliquityofUranushavebeenproposed.Kubo-Oka&Nakazawa(1995),investigated the tidalevolutionof satellite orbitsandexaminedthe possibility thatthe orbitaldecayof a retrogradesatellite leads to the large obliquityofUranus,butthelargemassrequiredforthehypotheticalsatellitemakesthispossibilityveryimplausible.Anasymmetric infallortorquesfromnearbymassconcentrationsduringthecollapseofthemolecularcloudcoreleadingtotheformationofthe SolarSystem,couldtwistthetotalangularmomentumvectoroftheplanetarysystem.Thistwistcouldgeneratetheobliquitiesof theouterplanets(Tremaine1991).Thismodelhasthedisadvantagesthattheouterplanetsmustformbeforetheinfalliscomplete andthattheconditionsfortheeventthatwouldproducethetwistareratherstrict.Themodelitselfisdifficulttobequantitatively tested.Tsiganisetal.(2005)proposedthatthecurrentorbitalarchitectureoftheouterSolarSystemcouldhavebeenproducedfrom aninitiallycompactconfigurationwithJupiterandSaturncrossingthe2:1orbitalresonancebydivergentmigration.Thecrossing led to close encountersamong the giant planets, producinglarge orbital eccentricities and inclinations which were subsequently dampedtothecurrentvaluebygravitationalinteractionswithplanetesimals.Theobliquitychangesduetothechangeintheorbital inclinations. Since the inclinations are damped by planetesimals interactions on timescales much shorter than the timescales for precessionduetothetorquesfromthe Sun,especiallyforUranusandNeptune,the obliquityreturnsto smallvaluesif itissmall beforetheencounters(Hoietal.2007). Large stochastic impacts at the last stage of the planetaryformationprocesshave been proposedas the possible cause of the planetaryobliquities(e.g.Safronov1969). Thelargeobliquityof Uranus(98 ) isusuallyattributedto a greattangentialcollision ◦ (GC)betweentheplanetandanEarth-sizeplanetesimaloccurredattheendoftheepochofaccretion(e.g.,Parisi&Brunini1997, Korycanskyetal.1990).ThecollisionimpartsanimpulsetoUranusandallowspreexistingsatellitesoftheplanettochangetheir orbits. Irregularson orbits with too large semimajor axis escape from the system (Parisi & Brunini 1997), while irregulars with a smallersemimajoraxismaybepushedto outerorinnerorbitsacquiringgreaterorlowereccentricitiesdependingonthe initial orbitalelements,thegeometryoftheimpactandthesatellitepositionatthemomentofimpact.Theorbitsexcitedbythisperturbation mustbeconsistentwiththepresentorbitalconfigurationoftheUranianirregulars(BP02). InanattempttoclarifytheoriginofUranusobliquityandofitsirregulars,weareusinginthisstudythemostupdatedinformation ontheirorbitalandphysicalproperties. InSection2,weimprovethemodeldevelopedinBP02forthefiveUranianirregularsknownatthatepochandextendourstudy tothenewfourUranianirregularsrecentlydiscoveredbyKavelaarsetal.(2004)andSheppardetal.(2005b).Theoriginofthese objectsaftertheGCisdiscussedinSection3,whereseveralmechanismsfortheoriginofProsperoareinvestigated.Thediscussion oftheresultsandtheconclusionsarepresentedinSection4. 2. Transferoftheirregularstotheircurrentorbits: AssumingtheGCscenario,thetransfersofthenineknownirregularstotheircurrentorbitsarecomputedfollowingtheprocedure developedinBP02forthefiveirregularsknownin2002.Wepresentimprovedcalculationsusingamorerealisticcodetocompute theevolutionoftheirregularscurrentorbitaleccentricities. IfthelargeobliquityofUranushasbeentheresultofagianttangentialimpact,theorbitsofpreexistingsatelliteschangeddue totheimpulseimpartedtotheplanetbythecollision.TheangularmomentumandimpulsetransfertotheUraniansystematimpact weremodeledusingtheUranuspresentdayrotationalandorbitalpropertiesasimputparameters(BP02). JustbeforetheGC,thesquareoftheorbitalvelocityν ofapreexistingsatelliteofnegligiblemassisgivenby: 1 2 1 ν2 =Gm , (1) 1 U r − a ! 1 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 3 rbeingthepositionofthesatelliteonitsorbitatthemomentoftheGC,a itsorbitalsemiaxisandm themassofUranusbefore 1 U the impact.The impactormassis m andG is the gravitationalconstant. Afterthe GC, the satellite is transferredto anotherorbit i withsemiaxisa acquiringthefollowingsquareofthevelocity: 2 2 1 ν2 =G(m +m) . (2) 2 U i r − a ! 2 Wesetν2 = A ν2 andν2 = B (1+m/m ) ν2,whereAandBarearbitrarycoefficients(0 < A 1, B > 0),ν beingtheescape 1 e 2 i U e ≤ e velocityatrbeforetheGC. Thesemiaxisofthesatelliteorbitbefore(a )andafter(a )theGCverifythefollowingsimplerelations: 1 2 r r a = , a = . (3) 1 2 2(1 A) 2(1 B) − − If A < Bthena < a .Inthespecialcaseof B = 1,theorbitsareunboundfromthesystem.If A > Bthena > a ,theinitial 1 2 1 2 orbitistransferredtoaninnerorbit.WhenA= B,theorbitalsemiaxisremainsunchanged(a =a ). 1 2 Thepositionrofthesatelliteonitsorbitattheepochoftheimpactmaybeexpressedinthefollowingform: 2G m B A r= U ′− 2, (4) (∆V)2 "√AcosΨ √(B A)+Acos2Ψ# ± ′− withB = B(1+m/m ).Sincestochasticprocessescanonlytakeplaceatverylatestagesinthehistoryofplanetaryaccretion(e.g. ′ i U Lissauer&Safronov1991),theGCisassumedtooccurattheendofUranusformation(e.g.Korycanskyetal.1990).Themassof UranusaftertheGC,(m +m ),istakenasUranus’presentmass.Ψistheanglebetweenν andtheorbitalvelocitychangeimparted i U 1 toUranus∆V.Ananalyticalexpressionfor∆VisderivedinBP02assumingthattheimpactisinelastic(Korycanskyetal.1990)as afunctionofm,theimpactparameterofthecollisionb,thepresentrotationangularvelocityofUranusΩ,thespinangularvelocity i whichUranuswouldhavetodayifthecollisionhadnotoccurredΩ ,andαwhichistheanglebetweenΩandΩ : 0 0 2R2 Ω2 2ΩΩ cosα 1/2 ∆V = U Ω2+ 0 0 , (5) 5b " (1+ mi)2(1+ mi )4 − (1+ mi)(1+ mi )2# mU 3mU mU 3mU R being the present equatorial radius of Uranus. A collision with the core itself was necessary in order to impart the required U additionalmassandangularmomentum(Korycanskyetal.1990).Sincebisanunknownquantity,wetakeitsmostprobablevalue: b=(2/3)R ,whereR isthecoreradiusofUranusatthemomentofcollisionassumedtobe1.8 104km(Korycanskyetal.1990, C C × Bodenheimer&Pollack1986).Theresultshavea smoothdependencewith theimpactormasswhichallowsustotakem 1m i (Parisi&Brunini1997). ∼ ⊕ Theminimumeccentricityoftheorbitsbeforethecollisionisgivenby: e =2(1 A) 1 if A 0.5 , e =1 2(1 A) if A>0.5, (6) 1min 1min − − ≤ − − whiletheminimumeccentricityoftheorbitsafterthecollisionis: e =2(1 B) 1 if B 0.5 , e =1 2(1 B) if B>0.5. (7) 2min 2min − − ≤ − − Theminimumpossiblevalueof∆V (∆V )isobtainedfromEq.(5)foraninitialperiodT =20hrs(T =2π/Ω )andα=70o min 0 0 0 (BP02).Therefore,althoughsimulationsofsolidaccretionproduceingeneralrandomspinorientations(e.g.Chambers2001),we further assume T = 20 hrs and α=70o in order to set a maximum boundon Eq. (4). Upper boundsin a (a ) and a (a ) are 0 1 1M 2 2M obtainedfromEq.(3)throughEqs.(4)and (5)taking∆V=∆V withΨ=180o,i.e.,assumingtheimpactinthedirectionopposite min totheorbitalmotionofthesatellitesandtakingthepositivesignofthesquarerootinEq.(4): G m (B A)2 G m (B A)2 a = U ′− , a = U ′− . (8) 1M 2M (∆V )2(1 A)(√B √A)2 (∆V )2(1 B)(√B √A)2 min − ′− min − ′− ForeachA,wecalculatethevalueofB(B= B/(1+m/m ))correspondingto thetransfertoa = a,whereaisthepresent ′ i U 2M orbitalsemiaxis of each one of the Uranian irregularsshown in the second columnof Table 1 in units of R . From Eq. (7), this U valueofBprovidestheminimumpossiblevalueofe ,e ,thattheorbitofeachirregularmayacquireatimpactforeachinitial 2min 2m conditionAandeveryinitialconditionforT ,α,Ψandm,i.e.,ifatransferofagivenorbit(A,B)isnotpossibleforΨ=180o,T = 0 i 0 20hrsandα= 70o,thesametransfer(A,B)isnotpossibleforanyotherincidentdirectionoftheimpactorandforanyothervalue ofT andαeither. 0 Since the orbits of the irregulars are time dependent, the orbital evolution of the five Uranian irregulars known in 2002 was computed in PB02 by numerical integration of the equations of the elliptical restricted three body problem formed by the Sun, Uranusandthesatellite.Inthispaper,wepresenttheorbitalevolutionofthenineknownUranianirregularsfor105 yrsusingthe 4 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV Satellite r [km] a[R ] e a[R ] e (∆a)/a) (∆e/e) s U mean i U i i i Caliban 49 283 0.191 287.5 0.1973 1.602x10 2 3.289x10 2 − − Sycorax 95 482 0.541 485 0.5436 6.307x10 3 4.795x10 3 − − Prospero 15 645 0.432 648 0.4342 4.585x10 3 4.997x10 3 − − Setebos 15 694 0.581 701.8 0.5853 1.123x10 2 7.366x10 3 − − Stephano 10 314 0.251 333 0.2781 6.058x10 2 0.1080 − Trinculo 5 336 0.218 361.6 0.2501 7.623x10 2 0.1475 − Ferdinand 6 813 0.660 839.3 0.6701 3.241x10 2 1.533x10 2 − − Francisco 6 169 0.142 280.6 0.3593 0.6604 1.530 Margaret 5.5 579 0.633 649 0.6705 0.1209 5.917x10 2 − Table 1. PresentparametersoftheUranianirregularsandorbitaldampingdueto gasdragexertedbyUranusextendedenvelope. r andaarethepresentphysicalradiusandthepresentorbitalsemiaxisoftheirregulars.e istheircalculatedmeaneccentricity s mean tabulatedinTable2.a ande aretheorbitalsemiaxisandeccentricityjustaftertheGC,while(∆a)/a)and(∆e/e)arethedamping i i i i oftheseorbitalelementssincetheepochoftheGCuntilthecontractionofUranusenvelope. Satellite e e e mean max min Caliban 0.191 0.315 0.072 Sycorax 0.514 0.594 0.438 Prospero 0.432 0.571 0.305 Setebos 0.581 0.704 0.463 Stephano 0.251 0.381 0.121 Trinculo 0.218 0.237 0.200 Ferdinand 0.660 0.970 0.393 Francisco 0.142 0.187 0.093 Margaret 0.633 0.854 0.430 Table2.VariationoftheeccentricityoftheUranianirregularsduetoSolarandgiantplanetperturbationsoveraperiodof105yrs. symplecticintegratorofWisdom&Holman(1991),wheretheperturbationsoftheSun,Jupiter,SaturnandNeptuneareincluded. Themean(e ),maximum(e )andminimum(e )eccentricitiesareshowninTable2foralltheknownUranianirregulars. mean max min Thetransferofeachsatellite fromitsoriginalorbittothepresentoneispossibleonlyforthosevaluesof(A,B)whichsatisfy theconditione <e .Asatellite didnotexistbeforetheimpactifithasnotransferandthesatelliteswiththewidestrangeof 2m max transfersarethosewiththehighestprobabilityofexistingbeforetheimpact. Thetransferswithinarangeof20R aroundeachpresentsatellitesemiaxis(a =a 20R ;atakenfromTable1)forallthe U 2M U ± Uranianirregularsare shown in Fig.1. Thereare few transfersforSetebos, Ferdinandand Margaret.Thismakesthe existenceof thesesatellitesbeforecollisionlittleprobable.TheonlytransfersforTrinculoandProsperoareclosetothepericenterofaneccentric initialouterorbit(e >0.58forProsperoande >0.62forTrinculo).Theminimumeccentricityaftercollisione forTrinculois 1m 1m 2m intherange[0.16-0.23],veryclosetoe (0.237).ThisresultgivesaverylowprobabilityfortheexistenceofTrinculobeforethe max GC.ForProsperoe isintherange[0.52-0.57],e e (0.571).Thereforethissatellite couldnotexistbeforetheGC.Ifthe 2m 2m max ∼ presentlargeobliquityofUranuswascausedbyalargeimpactattheendofitsformation,Prosperohadtooriginateaftertheevent. RelatingtheoriginoftheouterUraniansystemtoacommonformationprocess,alltheUranianirregularsprobablywereoriginated aftertheGC.Thepossiblepost-GCoriginofProsperoandtheotherUranianirregularsisdiscussedinthefollowingsection. 3. OriginofProsperoaftertheGreatCollision In this section, we analyze the possibility that Prospero have been capturedafter the GC. We investigatethe possible dissipative mechanisms able to produce its permanentcapture taking into account that the giant impact is assumed to have occurred at late stagesintheplanetaryaccretionprocess. 3.1.GasdragbyUranus’envelope Bodenheimer&Pollack(1986)andPollacketal.(1996)studiedtheformationofthegiantplanetsbyaccretionofsolidsandgas. Intheirmodel,thesocalledcoreinstabilityscenario,whenthemassofthecoreoftheplanethasgrownenoughagaseousenvelope beginsto formaroundit. For Uranus, its envelopeextendeduntilits accretionradiuswhich was 500R at the end of Uranus’ U ∼ formation (Bodenheimer& Pollack 1986). The formationof Uranus is completed when there is no more nebular gas to accrete. Otherwise, gas accretion by proto-Uranuswould have continued towards the runaway gas accretion phase and the planet would have now a massive gaseousenvelope.Bodenheimer& Pollack (1986) obtained that after the end of accretion, the radius of the envelopeofproto-Uranusremainedalmostconstant( 500R )overatimescaleof104 yrsandthencontractedrapidlyto 8R U U in105yrs.Thefinalcontractiontothepresent-daypla∼netaryradiusoccurredonaslowertimescaleof108yrs. ∼ Korycansky et al. (1990) carried out hydrodynamical calculations of the GC for a large set of initial conditions at the end of accretion. They found a sharp transition between the cases where almost all the mass of the envelope of Uranus remained after the impact and those where it was almost entirely dispersed by the impact. This implies that the impact should have not M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 5 Fig.1.ThetransferscapableofproducingthepresentorbitsoftheUranianIrregulars.A(B)isthesquareoftheratioofthesatellite’s speed just before(after)the impactto the escape velocityat the satellite’s locationjust before(after)the impact.e (e ) is the 1m 2m minimum eccentricity of the orbits before (after) collision. The full-black line A=B divides the upper region (the current orbits arise frominnerorbits)andthe lowerregion(thecurrentorbitsarise fromouterorbits).Thevalueofe tabulatedin Table2 is max shownonthedashedlineforcomparationwithe .Trinculo:emptydowntriangles,Caliban:fullcircles,Sycorax:emptyrhombus, 2m Ferdinand:emptysquares,Francisco:fulldowntriangles,Margaret:emptyuptriangles,Prospero:fulluptriangles,Setebos:empty hexagons,Stephano:emptycircles. dispersedtheenvelope,astherewouldhavebeennonebulargastore-accreteontheplanet.Theyshowedthattheenvelopereacts hydrodynamicallyatimpactanditexpandsoutward.Aftertheshockthegasfalls backonthe coreovera timescale offewhours beingthefinalresultareadjustmentinsteadofacatastrophictransformation.Thetimescaleforthishydrodynamicalprocessismuch shorterthantheorbitalperiodoftheirregularswhichisoftheorderofyears.WemaythenassumethattheGCdidnotchangethe envelopedensityprofile. 6 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV TheextendedenvelopeofUranuscouldbeinprinciple,asourceofgasallowingthecaptureofProsperoandtheotherirregulars aftertheGC.AssumingthattheGCdidnotchangetheenvelopeprofile,wefitfromFig.1ofKorycanskyetal.(1990)thedensity profile of Uranus’ gaseous envelope before the GC, ρ = c R 4 g cm 3 with c = 1036 and R being measured in cm. It gives a g te − − te nebulardensityof 4 10 13gcm 3attheboundaryof500R inagreementwiththeminimummassnebulamodel. − − U ∼ × As a firstapproximation,we computetheratio ofgasmasstraversedby a bodyofdensityρ andradiusr in a characteristic s s orbitalperiodP,tothemassofthebody.AssumingforthebodyacircularorbitofradiusR,wecalculatetheso-calledβparameter (Pollacketal.1979): P 2πRρ πr2 3πρ R β= = g s = g , (9) τ 43πρsr3s 2ρsrs whereτisthecharacteristictimescaleforchanginganyoftheorbitalparameters.Forthepermanentcapturetooccurβcannotbe verysmall,β 0.04(Pollacketal.1979).UsingEq.(9)andassuminganebuladensity10timesthatoftheminimumnebulamodel (c =1037)for≥anobjectthesizeofProspero(ρ =1.5gcm 3)andatProspero’spericenter(R=278R ),β 9 10 5,whichistoo te s − U − ∼ × smalltoaffecttheorbitofProspero. FollowingBP02,wenowinvestigatewithmoredetailthepossibleeffectofgasdragontheUranianirregularsaftertheGCdue toUranus’extendedenvelopebeforeitscontractiontoitspresentstate.FollowingtheprocedureofAdachietal.(1976),weobtain thetimevariationsoftheeccentricityeandsemiaxisaofeachUranianirregular.Thedragforceperunitmassisexpressedinthe form: C πr2 F = Cρ v2 ,C = D s, (10) − g rel 2m wherev istherelativevelocityofthesatellitewithrespecttothegas.Incomputingthesatellitemassm,asatellitemeandensity rel ρ of1.5gcm 3 istakenforallthesatellites(http://ssd.jpl.nasa.gov/?sat phys par).ThedragcoefficientC is 1andr iseach s − D s ∼ satelliteradiustakenforeachsatellitefromTable1. Assuming that the orbital elements are constant within one Keplerian period (the variations of a and e are very small), we considertheratesofchangeoftheelementsaveragedoveroneperiod,thatis: hddati=−Cπ(G(mi+mU)a)21 Z 2π ρg(e2(1++1e+co2seθc)o2sθ)32dθ 0 de = C(1 e2) G(mi+mU) 21 2π ρg(e+cosθ)(e2+1+2ecosθ)12dθ. (11) hdti −π − a ! Z (1+ecosθ)2 0 Sinceaftertheendofaccretionthegasdensityintheouterregionsoftheenvelopecontractsrapidly,wehaveintegratedEqs.(11) back in time on 104 yrsfor the 9 Uranianirregulars.We have taken v as the satellite orbitalvelocity since we have assumed a rel nullgasvelocity.Thisassumptionmaximizestheorbitaldampingforretrogradesatelliteswhichallowsustosetupperboundsin the damping effect for the orbital eccentricity and semiaxis of all the retrograde irregulars. The mean eccentricity e and the mean actual semiaxis a from Table 1 were taken as the initial conditions for the integrations. We take ρ = c R 4 g cm 3 with c = g te − − te 1036 and R = a(1 e2)/(1+ecosθ), a beingmeasured in cm. The orbitaldampingis shown in Table 1, where a and e are the i i − initialsemiaxisandeccentricityjustaftertheGCattheendofaccretionand∆a=(a a)and∆e=(e e ),arethedampingin i i mean − − theorbitalsemiaxisandeccentricity.Stephano,TrinculoandMargarethadexperiencedlittleorbitalevolutionwhileFranciscohad sufferedalargeorbitaldamping(seeTable1).ThepermittedtransfersofFig.1wouldincreaseforFranciscosincetheconditione 2m <e shouldbethensatisfied.However,theorbitalevolutionofthelargersatellites,inparticularofProspero,resultsnegligible.Even i increasingthenebuladensitybyafactorof10(c =1037)thedampingoftheorbitalelementsofProsperoistoosmallwith∆a/a= te i 0.046and∆e/e=0.048.ThissatellitehadnotexperiencedanyorbitalevolutionduetoUranus’extendedenvelopeandthencould i nothavebeencapturebygasdragaftertheGC. The pressure forces acting on a body traveling through the gas not only decelerates it, but also subjects it to stresses. If the stress isgreaterthan thestrengthof the body,the bodyisfracturedin fragmentsofdifferentsize. Thefragmentsmoveaway one anothersincedragforcesvaryinverselywith sizeandactto separatethem.Theaveragepressureontheforwardhemisphereofa non-rotating,sphericalbodyasitmovesthroughthegaswithrelativevelocityv isapproximatelyequaltothedynamicpressure, rel p = 1ρ v2 (Pollacketal.1979). Thebodywillfragmentinto piecesif p Q, where Q is thecompressivestrength.Values dyn 2 g rel dyn ≥ ofQontheorderof3 106 dynecm 2 areneededtoshatterstrong(e.g.rock/ice)targets(whichis10timeslowerthanthevalue − adoptedforasteroids),×whilecompressivestrengthontheorderof3 104dynecm 2areappropriateforrelativelyweak(snow-like) − × targets(Farinella&Davis1996,Stern1996).ForabodyoncircularorbitatthepresentpericenterofProspero(R=278R ),where U v isthecircularspeedaroundUranusatRandtakingc =1037forρ ,p 0.17dynecm 2,andatSycorax’spericenterp 1 rel te g dyn − dyn ∼ ∼ dynecm 2.Inbothcases p <<QandProsperocouldnothavebeenoriginatedbythedynamicalruptureofaparentobject. − dyn Acollisionmayfracturetheparentbodybutiftheenergyatimpactisnotsufficienttodispersethefragments,dragforcesmay acttoseparatethemagainsttheirmutualattraction.Therelativeimportanceoftheseeffectsismeasuredbytheratioǫ ofthedrag j forceonagivenfragmentjtothegravitationalforceactingonjbytheotherfragmentsi(Pollacketal.1979): F m Dj j ǫ = , (12) j F m Xi,i,j Gij j M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 7 whereF isthegravitationalforcebetweentheparticlesiandj,andF isthedragforceontheparticlejofradiusr : Gij Dj j C ρ πr2v2 F = D g j , (13) Dj 8πρ r3 3 s j Thefragmentjisdispersedbythegasifǫ >1(Pollacketal.1979).Inordertoestimatetheorderofmagnitudeoftheeffect,we j computedEq.(12)forj=ProsperoatProspero’spericentertakingintoaccountthegravitationalattractionduetoanotherfragmentof equalsize,andatSycoraxpericentertakingintoaccountthegravitationalforceofSycoraxonProspero.Inthefirstcase,weobtain ǫ =10 5andinthesecondcaseatSycoraxpericenterǫ =0.04.Wethenconcludethatpressureforcesarenotstrongenoughneither j − j tofragmentapossibleparentobjectfromwhichProsperooriginated,nortodisperseitsfragments.InadditionProsperosufferedno orbitalevolutionduetogasdragandcouldnothavebeencapturebyUranus’envelopeaftertheGC. 3.2.Pull-downcapture Within the GC scenario, runaway of the cores of the planets occurred during the first stages of accretion but stopped for each embryoafteritreachedasizeofabout1000Km.At10-35AUthefinalmassdistributioncontainedseveralhundredsofMars-size (or larger) bodies dominating the mass of the residual disk. Beauge´ et al. (2002), investigated the effects of the post-formation planetarymigrationonsatellitesorbits.Theyobtainedthatifthelarge-bodycomponent(composedofMars-sizebodies)dominated themassoftheresidualdisk,thepresentlyacceptedchangeintheorbitofUranusof 3AU istoolargeanditisnotcompatible ∼ withtheobserveddistributionofitssatellites.Evenanorbitalchangeof 1.5AUalreadycausessufficientinstabilitiestoejectall ∼ theUranianirregulars.Pull-downcapturecausedbytheorbitalexpansionoftheplanetcouldthennotbeaplausiblemechanismfor theoriginofProsperoandtheotherirregulars.Pull-downcapturecausedbythemassgrowthoftheplanetaftertheGCwouldnot bepossiblegiventheimpactisassumedtohaveoccurredattheendoftheaccretionprocesswhentherewas nomoremasstobe accretedbytheplanet. 3.3.Collisionlessinteractions Withintheframeworkoftherestrictedthree-bodyproblem,acaptureisalwaysfollowedbyanescape.Toendupwithalongterm capture,thesatellite hastodissipate energyinashorttime.Theentranceenergy∆E withinthe gravitationalfield oftheplanetis (Tsui1999): GM ∆E = 2.15µ2/3(1 δ) ⊙, (14) − − a p µ= M /M andδ<<1,whereM anda arethemassandorbitalsemiaxisoftheplanet. p p p Tsui (⊙1999), suggested a permanentcapture mechanism where a guest satellite encounterssome existing inner orbit massive planetary satellite causing its velocity vector to be deflected keeping the irregular in orbit around the planet. In this way, the effectivetwo-bodypotentialwouldbe abouttwice the entranceenergy∆E ofthe guestsatellite. TheradiusR ofthe orbitofthe 1 guestsatelliteafterdeflectionisthengivenby: 0.23 R = µ1/3a . (15) 1 1 δ p − In the case of Uranus,for a minimumentranceenergyof δ=0,the minimumpermanentorbitalradiusof the guestsatellite is R =955R .ThisvalueofR ismuchlargerthanthepresentsemiaxisofProspero(seeTable1),makingthecaptureofProsperoby 1 U 1 thismechanismimplausible. Thefactthatbinarieshaverecentlybeendiscoveredinnearlyallthesolarsystem’ssmall-bodyreservoirssuggeststhatbinary- planetgravitationalencounterscouldbring a possible mechanismfor irregularscapture (Agnor& Hamilton 2006). One possible outcomeofgravitationalencountersbetweenabinarysystemandaplanetisanexchangereaction,whereonememberofthebinary isexpelledandtheotherremainsboundtotheplanet.Tsui(1999)extendedthescenariooflargeanglesatellite-satellitescatteringto theformationofthePluto-CharonpairassumingthatPlutowasasatelliteofNeptuneandthatCharonwasaguestsatellite.Through Eqs.(14)and(15),theconditionsfortheescapeofthepairwasfound.Followingtheirscenario,letusconsiderthehypothesisthat ProsperowasamemberofaguestbinaryenteringUranus’field,withenergydensity∆E ,abovetheminimumdensitygivenby bin Eq.(14) and δ=0. A close encounter with Uranus could result in disruption of the binary, leading to the ejection of one member andcaptureoftheother.TheminimumsemiaxisR isgivenbyEq.(15).However,eventhisscenarioseemstobeunlikelysincethe 1 semiaxisofProsperoissmallerthan955R . U 3.4.Collisionalinteractions:Break-upprocesses Collisionalinteractionsbetweentwoplanetesimalspassingneartheplanetorbetweenaplanetesimalandaregularsatellite,theso calledbreak-upprocess,leadstotheformationofdynamicalgroupings(e.g.Colombo&Franklin1971,Nesvornyetal.2004).The resultingfragmentsofeachprogenitorbodyafterabreak-upwillformapopulationofirregularsexpectedtohavesimilarsurface composition,i.e.similarcolors,andirregularshapes,i.e.largetemporalvariationsinthelightcurveastheseirregularbodiesrotate. 8 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV Thecriticalrotationperiod(T )atwhichcentripetalaccelerationequalsgravitationalaccelerationforarotatingsphericalobject c is: 3π 1/2 T = , (16) c Gρ ! ob whereGisthegravitationalconstantandρ isthedensityoftheobject.Withρ =0.5,1,1.5and2gcm 3,T =4.7,3.3,2.7and2.3 ob ob − c hrs,respectively.TherotationperiodofProsperoisabout4hrs(Marisetal.2007a),anditseemsunlikelythatlightanddarksurface markingsonasphericalProsperocouldberesponsibleforitslightcurveamplitudeof0.2mag(Marisetal.2007a).Evenatlonger periods,realbodieswillsuffercentripetaldeformationintoasphericalshapes.Foragivendensityandspecificangularmomentum (H), the nature of the deformation depends on the strength of the object. In the limiting case of a strengthless (fluid) body, the equilibriumshapeshavebeenwellstudied(Chandrasekhar1987).ForH 0.304[inunitsof(GM3R )1/2],whereM(kg)isthe sphe ≤ mass of the objectand R is the radiusof an equal-volumesphere,the equilibriumshapes are the oblate Maclaurinspheroids. sphe For 0.304 H 0.390the equilibriumfiguresare triaxial Jacobiellipsoids. Strengthlessobjects with H > 0.390are rotationally ≤ ≤ unstabletofission.TheKuiperBeltobjects(KBOs),beingcomposedofsolidmatter,clearlycannotbestrengthless.However,itis likelythattheinteriorstructuresofthesebodieshavebeenrepeatedlyfracturedbyimpacts,andthattheirmechanicalresponseto appliedrotationalstressisapproximatelyfluid-like.Such“rubblepile”structurehaslongbeenstudiedintheasteroidbelt(Farinella et al.1981) and in the Kuiper Disk (Sheppard& Jewitt 2002, Jewitt & Sheppard 2002, Romanishin & Tegler 1999). Farinella & Davis(1996)obtainedthatKBOslargerthanabout100kmindiameteraremassiveenoughtosurvivecollisionaldisruptionover theageofthesolarsystem,butmayneverthelesshavebeeninternallyfracturedintorubblepiles. Whetheracollisionbetweenanimpactorandatargetresultsingrowthorerosiondependsprimarilyontheenergyoftheimpact and the mass and strength of the target. If the mass of the impactor is small compared to the mass of the target m , the energy s requiredatimpacttoresultinabreak-upisgivenby: 1 3Gm2 m v2 m S + s, (17) 2 s col ≥ s 5γR ms wherev is the collisionspeed, S is the impactstrength,R is the radiusofthe targetand γ is a parameterwhichspecifies the col ms fractionofcollisionalkineticenergythatgoesintofragmentkineticenergyandisestimatedtobe 0.1(Farinella&Davis1996). ∼ The speed of the fragments is critical when the target has a gravity field. Fragments moving slower than the local escape speed re-accumulatetoformrubberpilestructures. WeattempttoinvestigatewhetherProsperocouldbeacollisionalfragmentorifacollisiononaprimaryProsperowouldresult inarubberpilestructure. IncomputingEq.(17),v2 =v2+v2 ,wherev isthescapespeedatthetargetsurfaceandv isthetypicalapproachvelocity col e inf e inf ofthetwoobjectsatadistancelargecomparedwiththeHillsphereofthetarget.FortwobodiescollidingintheKuiperdisk,v is inf givenby(Lissauer&Stewart1993): 5 v2 =v2 e2+i2 , (18) inf k 4 ! wherev isthekeplerianvelocity,eisthemeanorbitaleccentricityandithemeanorbitalinclinationoftheKBOs.Wetake e =2 i k h i hi (Stern1996).Eq.(17)wascomputedusingEq.(18)forvaluesofeintherange[0.01-0.1]andorbitalsemiaxesoftheKBOsinthe range[30-60]AU.Incomputingthemassofthetargetm ,weconsidertheradiusR oftheKBOsintherange[10-500]kmand s ms densitiesbetween[0.5-2]gcm 3.Impactstrengthsintherange[3 104-3 106]ergcm 3weretaken.Weobtainthattargetswith − − × × R 210kmsufferdisruptionforallthevaluesoftheseparametersintheKuiperDisk.Prosperohasaradiusofjustonly15km.If ms ≤ ProsperooriginatedfromtheKuiperBelt,itwouldhavebeenmorelikelyacollisionalfragmentratherthanprimarybody.Prospero wouldpreservean irregularshape afterdisruptionsince it is suchsmall objectthat is unableto turnsphericalbecause its gravity cannotovercomematerialstrength.Prosperocouldhavebeencapturedduringthebreak-upeventifthetwoKBOscollidedwithin Uranus’Hillsphere,whichcouldbepossibleforaminimumorbitaleccentricityoftheoriginalKBOsof0.37. We alsoconsiderthecaseinwhichthetargetisasatelliteofUranuswhichcollideswithaKBO thatenterstheHillsphereof theplanet.Eq.(17)remainsvalidbutthefollowingexpressionofv isconsidered: inf v =v v , (19) inf ip sp − wherev isthevelocityoftheKB0withrespecttoUranusandv isthesatelliteorbitalvelocity,attheepochofthevent.Weassume ip sp thatthesatelliteorbitalvelocityiscircular.Inordertogetboundsintherelativevelocity,wetaketwovaluesofv ,v =v v . inf inf ip sp ± Forv ,weassumethattheKBOdescribesahyperbolicorbitaroundUranusduringtheapproachgiving: ip 2G(m +m ) GM v2 = i U + ⊙, (20) inf a a s kb whereweassumedGM /a astherelativevelocitybetweentheKBOandUranusfarfromtheencounter,(m +m )isthepresent kb i U massofUranusanda t⊙heorbitalsemiaxisofthesatellite.WecalculateEq.(17)usingEqs.(19)and(20)fora intherange[20,60] s kb AUanda [100-700]R forthesamevaluesofS,R anddensitieswehaveusedforthecollisionsamongKBOs.Weobtainthat s U ms M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 9 forallthepossibleparameters,anyUranus’satellitewithradiusR 1000kmsuffersdisruptionifitcollideswithaKBOlarger ms ≤ than10 km.Thisprocesswouldlead to the formationof two clustersofirregulars,one associatedto the preexistingsatellite and theothertotheprimaryKBO.Thisprocesshasthedisadvantagethatitisunlikelythatthepreexistingsatellitewereformedfroma circumplanetarydiskasregularsatellitesgiventhelargeorbitalsemiaxisrequiredforthisobject. Break-upprocessespredictorbitalclustering.However,noobviousdynamicalgroupingsareobservedattheirregularsofUranus. AfurtherintensivesearchofmorefaintirregularsaroundUranusisneededinordertolookfordynamicalandphysicalfamilies. 3.5.TheCGitselfasapossiblecapturemechanism WenowturntothequestionofwhethertheGCitselfcouldhaveprovidedacapturemechanism(BP02).Sinceallthetransferswith A> B leadtoamoreboundorbit,thisprocessmighttransformatemporarycaptureintoapermanentone(seeSection2andFig.1). ′ Moreover,apermanentcapturecouldevenoccurfromanheliocentricorbit(transferswithA=1). ItisinterestingtoestimatethenumberofobjectsN inheliocentricorbitsatthetimeoftheGC,atdistancesfromUranusless thanorequalto300R .AssumethattheGCoccurredwhenUranuswasalmostfullyformed,meaningthatitsfeedingzonewas U alreadydepleted of primordialplanetesimals. We assume that the objectspassing near Uranusat thattime were mainly escapees fromthe Kuiperbelt. Using the impactrate onto Uranusandthe distributionsofvelocitiesanddiametersgivenbyLevisonet al. (2000),andassumingthatthemassinthetransneptunianregionattheendoftheSolarSystemformationwas10timesitspresent mass,a back-of-the-envelopecalculationgivesoneobjectofdiameterD 20kmpassingatadistanceR 300R fromUranus U ≥ ≤ every6yrsattheendofaccretion(BP02).ThetypicalcrossingtimeT amongprotoplanetsintheouterSolarSystemislargerthan C onemillonyears(Zhouetal.2007).ThenumberofobjectspassingnearUranusduringatimescaleT isthen167000,whichgives C a probabilityof 6 10 6 for the capture of an object at about 300 R by the GC. This low rate of incoming objects, makes the − U × possibilityofthecaptureofalltheirregularsfromheliocentricorbitsdifficult.Eventhecaptureofasingleobject,Prospero(note thatProsperocouldnothaveanorbitboundtotheplanetbeforetheGC),turnsouttobelowprobable.Sincetemporarycapturecan lengthenthetimewhichapassingbodycanspendneartheplanet,amoreplausiblesituationarisesifweassumethattheGCcould produce the permanent capture of one or more parent objects which were orbiting temporarily around Uranus being the present irregularstheresultofacollisionalbreak-upoccurringaftertheGC. 4. DiscussionandConclusions ItisusuallybelievedthatthelargeobliquityofUranusistheresultofagreattangentialcollision(GC)withanEarth-sizedproto- planetattheendofthe accretionprocess.We havecalculatedthetransferofangularmomentumandimpulseatimpactandhave shownthatthe GChadstronglyaffectedtheorbitsofUraniansatellites. We calculatethetransferoftheorbitsofthenineknown UranianirregularsbytheGC.Veryfewtransfersexistforfiveofthenineirregulars,makingtheirexistencebeforetheGChardly expected.Inparticular,Prosperocouldnotexistatthetime oftheGC. Then,eitherProsperohadto originateaftertheGC orthe GCdidnotoccur,inwhichcaseanothertheoryabletoexplainUranus’obliquityandtheformationoftheUranianregularsatellites wouldbeneeded.ItisusuallybelievedthattheregularsatellitesofUranushaveaccretedfrommaterialplacedintoorbitbytheGC (Stevensonetal.1986). WithintheGCscenario,severalpossiblemechanismsforthecaptureofProsperoaftertheGCwereinvestigated.IftheUranian irregulars belong to individual captures and relating the origin of the outer uranian system to a common formation process, gas dragby Uranus’envelopeand pull-downcaptureseem to be implausible.Three-bodygravitationalencountersmightbe a source of permanentcapture. However,we foundthat the minimum permanentorbitalradiusof a guestsatellite of Uranusis 955 R U ∼ whilethecurrentsemiaxisofProsperois645R .TheGCitselfcouldprovideamechanismofpermanentcaptureandthecapture U ofProsperocouldhaveoccurredfromaheliocentricorbitasisrequiredwithintheGCscenario,butduetothelowrateofincoming objectsit turnsoutto bedifficult.Break-upprocessescouldbe themechanismforthe originofProsperoandthe otherirregulars in the frame of differentscenarios. Prospero might be a fragmentof a primary KBO fracturedby a collision with anotherKBO. ThefragmentcouldhavebeencapturedbyUranusifthetwoKBOshadaminimumorbitaleccentricityof0.37.Prosperocouldbe a secondarymemberofa collisionalfamilyoriginatedbythecollisionbetweenanothersatellite ofUranusanda KBO wherethe parentsatelliteofProsperocouldhavebeencapturedbyanymechanismbeforeoraftertheGC.Thisprocesshasthedisadvantage thatitisunlikelythatthepreexistingsatellitewereformedfromacircumplanetarydiskasregularsatellitesgiventhelargeorbital semiaxis required for this object. Since collisional scenarios require in general high collision rates, perhaps the irregulars were originally much more numerousthan now. Then, Prospero and also the other irregulars might be the result of mutual collisions amonghypotheticalpreexistingirregulars(Nesvornyetal. 2003, 2007)whichcouldhavebeencapturedbyanyothermechanism beforetheGC. TheknowledgeofthesizeandshapedistributionofirregularsisimportanttoknowtheirrelationtotheprecursorKuiperBelt population.It could bring valuable clues to investigate if they are collisional fragmentsfrom break-upprocesses occuring at the Kuiper Belt and thus has nothing to do with how they were individually captured later by the planet, or if they are collisional fragmentsproducedduringorafterthecaptureevent(Nesvornyetal.2003,2007).ThedifferentialsizedistributionoftheUranian irregularsapproximatesapowerlawwithanexponentq=1.8(Sheppardetal.2005b).Ifweassumethatthesizedistributionofthe nineirregularswithradiigreaterthan7kmextendsdowntoradiiofabout1km,wewouldexpectabout75irregularsofthissizeor larger(Sheppardetal.2005b). ThenucleiofJupiterfamilycometsarewidelyconsideredtobekilometer-sizedfragmentsproducedcollisionallyintheKuiper Belt(Farinella&Davis1996).Jewittetal.(2003)comparedtheshapedistributionofcometarynucleiintheJupiterfamilywiththe shapedistributionofsmallmain-beltasteroidsofsimilarsize(1km-10km)andwiththeshapedistributionoffragmentsproduced 10 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV in laboratoryimpactexperiments.They foundthat while the asteroidsand laboratoryimpactfragmentsshow similar distribution of axis ratio ( b/a 0.7),cometarynucleiare moreelongated( b/a 0.6).Theypredictthatif cometsreflecttheir collisional h i∼ h i∼ originintheKuiperBeltfollowedbysublimation-drivenmasslossonceinsidetheorbitofJupiter,smallKBOsshouldhaveaverage shapesconsistentwith those ofcollisionally producedfragments(i.e., b/a 0.7).To date, constraintson the shapesof onlythe h i∼ largestKBOsareavailable.Prosperobeingabitlargerthancometarynuclei,displaysavariabilityof0.21magintheRband(Maris etal.2007a).Thiscorrespondstoanaxisratioprojectedintotheplaneofthesky,b/aof0.8.Theknowledgeofthesizeandshape distribution of irregulars would shed light in the size and shape distribution of small KBOs as well as on the irregulars capture mechanism. Colors are an important diagnostic tool in attempting to unveil the physical status and the origin of the Uranian irregulars. In particular it would be interesting to assess whether it is possible to define subclasses of irregulars just looking at colors, and comparingcolorsofthesebodieswithcolorsofminorbodiesintheouterSolarSystem.Avalilableliteraturedatashowadispersion in thepublishedvalueslargerthanquotederrorsforeachUranianirregular(Mariset al.2007a, andreferencestherein).We have concludedinMarisetal.(2007a),thattheUranianirregularsareslightlyredbuttheyarenotasredasthereddestKBOs. An intensivesearchforfainter irregularsanda longterm programof observationsableto recoverin a self consistentmanner light-curves,colorsandphaseeffectsinformationsismandatory. Acknowledgements. MGPresearchwassupportedbyInstitutoArgentinodeRadioastronom´ıa,IAR-CONICET,ArgentinaandbyCentrodeAstrof´ısica,Fondode Investigacio´navanzadoenAreasPrioritarias,FONDAPnumber15010003,Chile.MMacknowledgesFONDAPforfinantialsupportduringavisittoUniversidad deChile.PartoftheworkofMMhasbeensupportedbyINAFFFO-FondoRicercaLibera-2006.ABresearchwassupportedbyIALP-CONICET.Weappreciate theusefulsuggestionsbytheReviewer,whichhavehelpedustogreatlyimprovethispaper. 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